Time-domain computation of scattering spectra for use in spectroscopic metrology

Abstract

In an embodiment, a method of time-domain simulation for simulating scattering spectra is described. The method may provide for computing spatial derivatives including computing spectral derivatives in some portion of the domain and finite difference derivatives in some other portion of the domain; forming an equation for non-reflecting boundary conditions; and computing a time-stepping scheme using a high-order unconditionally stable computation method.

1. A method of time-domain computation for simulating electromagnetic (EM) scattering spectra comprising:
 a processor computing spatial derivatives comprising computing spectral derivatives in some portion of the domain and finite difference derivatives in some other portion of the domain; 
 the processor forming an equation for boundary conditions; and 
 the processor computing a time-stepping scheme. 
 
   
   
     2. The method of  claim 1 , further comprising computing spatial derivatives of a transition portion of the domain using a linear combination of the spectral derivatives and finite difference derivatives. 
   
   
     3. The method of  claim 1 , wherein forming the equation for boundary conditions comprises forming the equation for non-reflecting boundary conditions to truncate the computational domain. 
   
   
     4. The method of  claim 3 , further comprising adding an absorbing layer to the computational domain. 
   
   
     5. The method of  claim 3 , wherein forming the equation for non-reflecting boundary conditions comprises computing a solution to a wave equation. 
   
   
     6. The method of  claim 1 , wherein computing the time-stepping scheme comprises computing the time-stepping using a high-order unconditionally stable computation method. 
   
   
     7. The method of  claim 1 , wherein materials under simulation have frequency-dependent dispersive dielectric properties. 
   
   
     8. The method of  claim 1 , wherein the time-stepping scheme comprises a field transformed system of equations for oblique incidence on a periodic domain. 
   
   
     9. The method of  claim 1 , further comprising computing a linear combination of plane waves for an incident pulse to simulate a finite aperture for the incident pulse in a single time-domain solution. 
   
   
     10. A method of time-domain computation for simulating electromagnetic (EM) scattering spectra comprising:
 a processor computing spatial derivatives of the domain; 
 the processor forming an equation for non-reflecting boundary conditions to truncate the computational domain; and 
 the processor computing a time-stepping scheme. 
 
   
   
     11. The method of  claim 10 , wherein computing spatial derivatives of the domain comprises computing spectral derivatives in some portion of the domain and finite difference derivatives in some other portion of the domain. 
   
   
     12. The method of  claim 10 , further comprising computing spatial derivatives of a transition portion of the domain using a linear combination of the spectral derivatives and finite difference derivatives. 
   
   
     13. The method of  claim 10 , wherein forming the equation for non-reflecting boundary conditions comprises computing a solution to a wave equation. 
   
   
     14. The method of  claim 10 , wherein computing the time-stepping scheme comprises computing the time-stepping using a high-order unconditionally stable computation method. 
   
   
     15. The method of  claim 10 , wherein materials under simulation comprise frequency-dependent dispersive dielectric properties. 
   
   
     16. The method of  claim 10 , wherein the time-stepping scheme comprises a field transformed system of equations for oblique incidence on a periodic domain. 
   
   
     17. A method of time-domain computation for simulating electromagnetic (EM) scattering spectra comprising:
 a processor computing spatial derivatives of the domain; 
 the processor forming an equation for boundary conditions; and 
 the processor computing a time-stepping scheme using a high-order unconditionally stable computation method. 
 
   
   
     18. The method of  claim 17 , wherein computing spatial derivatives of the domain comprises computing spectral derivatives in some portion of the domain and finite difference derivatives in some other portion of the domain. 
   
   
     19. The method of  claim 17 , further comprising computing spatial derivatives of a transition portion of the domain using a linear combination of the spectral derivatives and finite difference derivatives. 
   
   
     20. The method of  claim 17 , wherein forming the equation for boundary conditions comprises forming the equation for non-reflecting boundary conditions to truncate the computational domain. 
   
   
     21. The method of  claim 20 , wherein forming the equation for non-reflecting boundary conditions comprises computing a solution to a wave equation. 
   
   
     22. A system for measuring electromagnetic (EM) spectra and computing a time-domain simulation for simulating scattering spectra comprising:
 a broadband EM radiation source for illuminating a wafer; 
 a spectrometer to detect different spectral components of radiation reflected from the wafer; and 
 a processor to compute scattering spectra by:
 computing spatial derivatives comprising computing spectral derivatives in some portion of the domain and finite difference derivatives in some other portion of the domain; 
 forming an equation for non-reflecting boundary conditions; 
 computing a high-order unconditionally stable computation method; 
 varying the model geometry; and 
 re-computing the computed scattering spectra until the computed scatting spectra matches a measured spectrum. 
 
 
   
   
     23. The system of  claim 22 , wherein the processor computes scattering spectra by computing spatial derivatives of a transition portion of the domain using a linear combination of the spectral derivatives and finite difference derivatives. 
   
   
     24. The system of  claim 22 , wherein materials under simulation comprise frequency-dependent dispersive dielectric properties. 
   
   
     25. The system of  claim 22 , wherein the processor computes scattering spectra by computing spatial derivatives of a transition portion of the time-domain using a linear combination of the spectral derivatives and finite difference derivatives. 
   
   
     26. The system of  claim 22 , wherein the time-domain under computation comprises a time-domain of finite extent. 
   
   
     27. The system of  claim 22 , wherein the time-domain under computation comprises a periodic domain. 
   
   
     28. The system of  claim 22 , wherein the processor computes a field transformed system of equations for oblique incidence on a periodic domain.